Asymptotic ruin probabilities for a renewal risk model with a random number of delayed claims

Jinzhu Li

doi:10.3934/jimo.2022112

Article Contents

2023, Volume 19, Issue 6: 3840-3853. Doi: 10.3934/jimo.2022112

This issue Previous Article Next Article Interval estimation methods of fault estimation for discrete-time switched systems

Asymptotic ruin probabilities for a renewal risk model with a random number of delayed claims

Jinzhu Li^,

School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, China

^*Corresponding author: Jinzhu Li
^*Corresponding author: Jinzhu Li

Received: November 2021

Revised: May 2022

Early access: July 2022

Published: June 2023

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Abstract

In this paper, we consider a non-standard renewal risk model, in which each main claim may produce a random number of delayed claims. Such a model takes full account of lasting impacts of severe claims and hence may avoid the underestimation of an insurer's operation risk within a relatively long period. We obtain asymptotic expansions for the finite-time ruin probability, given that the claim size distributions belong to the general subexponential class. When the claim size distributions are restricted to the smaller class of extended regular variation, we push our study forward to the infinite-time ruin probability.

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Mathematics Subject Classification: Primary: 62P05; Secondary: 62E20, 91B30.

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References

[1]	K. B. Athreya and P. E. Ney, Branching Processes, Springer-Verlag, New York-Heidelberg, 1972.
[2]	N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.
[3]	Y. Chen and K. W. Ng, The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims, Insurance Math. Econom., 40 (2007), 415-423. doi: 10.1016/j.insmatheco.2006.06.004.
[4]	Y. Chen and K. C. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stoch. Models, 25 (2009), 76-89. doi: 10.1080/15326340802641006.
[5]	D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stochastic Process. Appl., 49 (1994), 75-98. doi: 10.1016/0304-4149(94)90113-9.
[6]	P. Embrechts, C. Klüuppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, New York, 1997. doi: 10.1007/978-3-642-33483-2.
[7]	K. Fu and H. Li, Asymptotic ruin probability of a renewal risk model with dependent by-claims and stochastic returns, J. Comput. Appl. Math., 306 (2016), 154-165. doi: 10.1016/j.cam.2016.03.038.
[8]	K. Fu, Y. Qiu and A. Wang, Estimates for the ruin probability of a time-dependent renewal risk model with dependent by-claims, Appl. Math. J. Chinese Univ. Ser. B, 30 (2015), 347-360. doi: 10.1007/s11766-015-3297-4.
[9]	Q. Gao, J. Zhuang and Z. Huang, Asymptotics for a delay-claim risk model with diffusion, dependence structures and constant force of interest, J. Comput. Appl. Math., 353 (2019), 219-231. doi: 10.1016/j.cam.2018.12.036.
[10]	X. Hao and Q. Tang, A uniform asymptotic estimate for discounted aggregate claims with subexponential tails, Insurance Math. Econom., 43 (2008), 116-120. doi: 10.1016/j.insmatheco.2008.03.009.
[11]	J. Li, On pairwise quasi-asymptotically independent random variables and their applications, Statist. Probab. Lett., 83 (2013), 2081-2087. doi: 10.1016/j.spl.2013.05.023.
[12]	J. Li, On the joint tail behavior of randomly weighted sums of heavy-tailed random variables, J. Multivariate Anal., 164 (2018), 40-53. doi: 10.1016/j.jmva.2017.10.008.
[13]	A. J. McNeil, R. Frey and P. Embrechts, Quantitative Risk Management. Concepts, Techniques and Tools, Princeton University Press, Princeton, NJ, 2005.
[14]	Q. Tang and Z. Yuan, Randomly weighted sums of subexponential random variables with application to capital allocation, Extremes, 17 (2014), 467-493. doi: 10.1007/s10687-014-0191-z.
[15]	D. Wang and Q. Tang, Tail probabilities of randomly weighted sums of random variables with dominated variation, Stoch. Models, 22 (2006), 253-272. doi: 10.1080/15326340600649029.
[16]	H. R. Waters and A. Papatriandafylou, Ruin probabilities allowing for delay in claims settlement, Insurance Math. Econom., 4 (1985), 113-122. doi: 10.1016/0167-6687(85)90005-8.
[17]	Y. Xiao and J. Guo, The compound binomial risk model with time-correlated claims, Insurance Math. Econom., 41 (2007), 124-133. doi: 10.1016/j.insmatheco.2006.10.009.
[18]	H. Yang and J. Li, On asymptotic finite-time ruin probability of a renewal risk model with subexponential main claims and delayed claims, Statist. Probab. Lett., 149 (2019), 153-159. doi: 10.1016/j.spl.2019.01.037.
[19]	K. C. Yuen and J. Guo, Ruin probabilities for time-correlated claims in the compound binomial model, Insurance Math. Econom., 29 (2001), 47-57. doi: 10.1016/S0167-6687(01)00071-3.
[20]	K. C. Yuen, J. Guo and K. W. Ng, On ultimate ruin in a delayed-claims risk model, J. Appl. Probab., 42 (2005), 163-174. doi: 10.1239/jap/1110381378.
[21]	Y. Zhang, X. Shen and C. Weng, Approximation of the tail probability of randomly weighted sums and applications, Stochastic Process. Appl., 119 (2009), 655-675. doi: 10.1016/j.spa.2008.03.004.